27 results found
Mishra S, Flaxman S, Berah T, et al., 2022, πVAE: a stochastic process prior for Bayesian deep learning with MCMC, Statistics and Computing, Vol: 32, ISSN: 0960-3174
Stochastic processes provide a mathematically elegant way to model complex data. In theory, they provide flexible priors over function classes that can encode a wide range of interesting assumptions. However, in practice efficient inference by optimisation or marginalisation is difficult, a problem further exacerbated with big data and high dimensional input spaces. We propose a novel variational autoencoder (VAE) called the prior encoding variational autoencoder (πVAE). πVAE is a new continuous stochastic process. We use πVAE to learn low dimensional embeddings of function classes by combining a trainable feature mapping with generative model using a VAE. We show that our framework can accurately learn expressive function classes such as Gaussian processes, but also properties of functions such as their integrals. For popular tasks, such as spatial interpolation, πVAE achieves state-of-the-art performance both in terms of accuracy and computational efficiency. Perhaps most usefully, we demonstrate an elegant and scalable means of performing fully Bayesian inference for stochastic processes within probabilistic programming languages such as Stan.
Murray P, Wood B, Buehler H, et al., 2022, Deep hedging: continuous reinforcement learning for hedging of general portfolios across multiple risk aversions, 3rd ACM International Conference on AI in Finance (ICAIF ’22), Publisher: ACM
We present a method for finding optimal hedging policies for arbitrary initial portfolios and market states. We develop a novel actor-critic algorithm for solving general risk-averse stochastic control problems and use it to learn hedging strategies across multiple risk aversion levels simultaneously. Wedemonstrate the effectiveness of the approach with a numerical example in a stochastic volatility environment.
Bolko AE, Christensen K, Pakkanen MS, et al., 2022, A GMM approach to estimate the roughness of stochastic volatility, Journal of Econometrics, ISSN: 0304-4076
We develop a GMM approach for estimation of log-normal stochastic volatility modelsdriven by a fractional Brownian motion with unrestricted Hurst exponent. We show thata parameter estimator based on the integrated variance is consistent and, under strongerconditions, asymptotically normally distributed. We inspect the behavior of our procedurewhen integrated variance is replaced with a noisy measure of volatility calculated from discretehigh-frequency data. The realized estimator contains sampling error, which skews the fractalcoefficient toward “illusive roughness.” We construct an analytical approach to control theimpact of measurement error without introducing nuisance parameters. In a simulation study,we demonstrate convincing small sample properties of our approach based both on integratedand realized variance over the entire memory spectrum. We show the bias correction attenuatesany systematic deviance in the parameter estimates. Our procedure is applied to empiricalhigh-frequency data from numerous leading equity indexes. With our robust approach theHurst index is estimated around 0.05, confirming roughness in stochastic volatility.
Morariu-Patrichi M, Pakkanen MS, 2022, State-dependent Hawkes processes and their application to limit order book modelling, Quantitative Finance, Vol: 22, Pages: 563-583, ISSN: 1469-7688
We study statistical aspects of state-dependent Hawkes processes, which are an extension of Hawkesprocesses where a self- and cross-exciting counting process and a state process are fully coupled, interacting with each other. The excitation kernel of the counting process depends on the state process that,reciprocally, switches state when there is an event in the counting process. We first establish the existenceand uniqueness of state-dependent Hawkes processes and explain how they can be simulated. Then wedevelop maximum likelihood estimation methodology for parametric specifications of the process. Weapply state-dependent Hawkes processes to high-frequency limit order book data, allowing us to builda novel model that captures the feedback loop between the order flow and the shape of the limit orderbook. We estimate two specifications of the model, using the bid–ask spread and the queue imbalanceas state variables, and find that excitation effects in the order flow are strongly state-dependent. Additionally, we find that the endogeneity of the order flow, measured by the magnitude of excitation, is alsostate-dependent, being more pronounced in disequilibrium states of the limit order book.
Buehler H, Murray P, Pakkanen MS, et al., 2022, Deep hedging: learning to remove the drift, Risk, ISSN: 1743-9477
We present a machine learning approach for finding minimal equivalent martingale measures for markets simulatorsof tradable instruments, e.g. for a spot price and optionswritten on the same underlying. We extend our resultsto markets with frictions, in which case we find “nearmartingale measures” under which the prices of hedginginstruments are martingales within their bid/ask spread.By removing the drift, we are then able to learn using DeepHedging a “clean” hedge for an exotic payoff which is notpolluted by the trading strategy trying to make moneyfrom statistical arbitrage opportunities. We correspondingly highlight the robustness of this hedge vs estimationerror of the original market simulator. We discuss applications to two market simulators.
In this work we derive limit theorems for trawl processes. First, we study the asymptotic behaviorof the partial sums of the discretized trawl process (Xi∆n)bntc−1i=0 , under the assumption that as n ↑ ∞,∆n ↓ 0 and n∆n → µ ∈ [0, +∞]. Second, we prove a general result on functional convergence indistribution of trawl processes. As an application of this result, we show that a trawl process whoseL´evy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian movingaverage process.
Bennedsen M, Lunde A, Pakkanen MS, 2021, Decoupling the short- and long-term behavior of stochastic volatility, Journal of Financial Econometrics, ISSN: 1479-8409
We introduce a new class of continuous-time models of the stochastic volatility of asset prices. The models can simultaneously incorporate roughness and slowly decaying autocorrelations, including proper long memory, which are two stylized facts often found in volatility data. Our prime model is based on the so-called Brownian semistationary process and we derive a number of theoretical properties of this process, relevant to volatility modeling. Applying the models to realized volatility measures covering a vast panel of assets, we find evidence consistent with the hypothesis that time series of realized measures of volatility are both rough and very persistent. Lastly, we illustrate the utility of the models in an extensive forecasting study; we find that the models proposed in this paper outperform a wide array of benchmarks considerably, indicating that it pays off to exploit both roughness and persistence in volatility forecasting.
Heinrich C, Pakkanen MS, Veraart AED, 2019, Hybrid simulation scheme for volatility modulated moving average fields, Mathematics and Computers in Simulation, Vol: 166, Pages: 224-244, ISSN: 0378-4754
Jacquier A, Pakkanen MS, Stone H, 2018, Pathwise large deviations for the rough Bergomi model, Journal of Applied Probability, Vol: 55, Pages: 1078-1092, ISSN: 0021-9002
Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.
McCrickerd R, Pakkanen MS, 2018, Turbocharging Monte Carlo pricing for the rough Bergomi model, Quantitative Finance, Vol: 18, Pages: 1877-1886, ISSN: 1469-7688
The rough Bergomi model, introduced by Bayer, Friz and Gatheral [Quant.Finance 16(6), 887-904, 2016], is one of the recent rough volatility modelsthat are consistent with the stylised fact of implied volatility surfaces beingessentially time-invariant, and are able to capture the term structure of skewobserved in equity markets. In the absence of analytical European optionpricing methods for the model, we focus on reducing the runtime-adjustedvariance of Monte Carlo implied volatilities, thereby contributing to themodel's calibration by simulation. We employ a novel composition of variancereduction methods, immediately applicable to any conditionally log-normalstochastic volatility model. Assuming one targets implied volatility estimateswith a given degree of confidence, thus calibration RMSE, the results wedemonstrate equate to significant runtime reductions - roughly 20 times onaverage, across different correlation regimes.
Morariu-Patrichi M, Pakkanen MS, 2018, Hybrid Marked Point Processes: Characterization, Existence and Uniqueness, Market Microstructure and Liquidity, Vol: 04, Pages: 1950007-1950007, ISSN: 2382-6266
<jats:p> In this paper, we introduce a class of hybrid marked point processes, which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process interacting with its state process. We show the existence and uniqueness of hybrid marked point processes under general assumptions, extending the results of Massoulié (1998) on interacting point processes. </jats:p>
Bennedsen M, Lunde A, Pakkanen MS, 2017, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics, Vol: 21, Pages: 931-965, ISSN: 0949-2984
We introduce a simulation scheme for Brownian semistationary processes, whichis based on discretizing the stochastic integral representation of the processin the time domain. We assume that the kernel function of the process isregularly varying at zero. The novel feature of the scheme is to approximatethe kernel function by a power function near zero and by a step functionelsewhere. The resulting approximation of the process is a combination ofWiener integrals of the power function and a Riemann sum, which is why we callthis method a hybrid scheme. Our main theoretical result describes theasymptotics of the mean square error of the hybrid scheme and we observe thatthe scheme leads to a substantial improvement of accuracy compared to theordinary forward Riemann-sum scheme, while having the same computationalcomplexity. We exemplify the use of the hybrid scheme by two numericalexperiments, where we examine the finite-sample properties of an estimator ofthe roughness parameter of a Brownian semistationary process and study MonteCarlo option pricing in the rough Bergomi model of Bayer et al. (2015),respectively.
Pakkanen MS, Sottinen T, Yazigi A, 2017, On the conditional small ball property of multivariate Lévy-driven moving average processes, Stochastic Processes and their Applications, Vol: 127, Pages: 749-782, ISSN: 0304-4149
We study whether a multivariate Lévy-driven moving average process can shadow arbitrarily closely any continuous path, starting from the present value of the process, with positive conditional probability, which we call the conditional small ball property. Our main results establish the conditional small ball property for Lévy-driven moving average processes under natural non-degeneracy conditions on the kernel function of the process and on the driving Lévy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional Lévy processes and multivariate Lévy-driven Ornstein–Uhlenbeck processes.
Lukkarinen J, Pakkanen MS, 2016, Arbitrage without borrowing or short selling?, Mathematics and Financial Economics, Vol: 11, Pages: 263-274, ISSN: 1862-9679
We show that a trader, who starts with no initial wealth and is not allowedto borrow money or short sell assets, is theoretically able to attain positivewealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwiseRiemann-Stieltjes integral. Our result exemplifies the potential intricacies offormulating economically meaningful self-financing conditions in continuoustime, when one leaves the conventional arbitrage-free framework.
Pakkanen MS, Réveillac A, 2016, Functional limit theorems for generalized variations of the fractional Brownian sheet, Bernoulli, Vol: 22, Pages: 1671-1708, ISSN: 1350-7265
We prove functional central and non-central limit theorems for generalizedvariations of the anisotropic d-parameter fractional Brownian sheet (fBs) forany natural number d. Whether the central or the non-central limit theoremapplies depends on the Hermite rank of the variation functional and on thesmallest component of the Hurst parameter vector of the fBs. The limitingprocess in the former result is another fBs, independent of the original fBs,whereas the limit given by the latter result is an Hermite sheet, which isdriven by the same white noise as the original fBs. As an application, wederive functional limit theorems for power variations of the fBs and discusswhat is a proper way to interpolate them to ensure functional convergence.
Bender C, Pakkanen MS, Sayit H, 2015, Sticky continuous processes have consistent price systems, Journal of Applied Probability, Vol: 52, Pages: 586-594, ISSN: 1475-6072
Under proportional transaction costs, a price process is said to have aconsistent price system, if there is a semimartingale with an equivalentmartingale measure that evolves within the bid-ask spread. We show that acontinuous, multi-asset price process has a consistent price system, underarbitrarily small proportional transaction costs, if it satisfies a naturalmulti-dimensional generalization of the stickiness condition introduced byGuasoni [Math. Finance 16(3), 569-582 (2006)].
Barndorff-Nielsen OE, Pakkanen MS, Schmiegel J, 2014, Assessing Relative Volatility/Intermittency/Energy Dissipation, Electronic Journal of Statistics, Vol: 8, Pages: 1996-2021, ISSN: 1935-7524
We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence.
Lukkarinen J, Pakkanen MS, 2014, ON THE POSITIVITY OF RIEMANN-STIELTJES INTEGRALS (vol 87, pg 400, 2013), BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, Vol: 89, Pages: 524-524, ISSN: 0004-9727
Pakkanen MS, 2014, Limit theorems for power variations of ambit fields driven by white noise, Stochastic Processes and Their Applications, Vol: 124, Pages: 1942-1973, ISSN: 0304-4149
We study the asymptotics of lattice power variations of two-parameter ambit fields driven by white noise. Our first result is a law of large numbers for power variations. Under a constraint on the memory of the ambit field, normalized power variations converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This result holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asymptotic behavior. Our second result is a stable central limit theorem for thinned power variations. © 2014 Elsevier B.V. All rights reserved.
Bayraktar E, Pakkanen MS, Sayit H, 2014, On the Existence Of Consistent Price Systems, Stochastic Analysis and Applications, Vol: 32, Pages: 152-162, ISSN: 0736-2994
Corcuera JM, Hedevang E, Pakkanen MS, et al., 2013, Asymptotic theory for Brownian semi-stationary processes with application to turbulence, STOCHASTIC PROCESSES AND THEIR APPLICATIONS, Vol: 123, Pages: 2552-2574, ISSN: 0304-4149
Lukkarinen J, Pakkanen MS, 2013, On the positivity of Riemann–Stieltjes integrals, Bulletin of the Australian Mathematical Society, Vol: 87, Pages: 400-405, ISSN: 0004-9727
Lappi E, Pakkanen MS, Akesson B, 2012, AN APPROXIMATIVE METHOD OF SIMULATING A DUEL, Winter Simulation Conference (WSC), Publisher: IEEE, ISSN: 0891-7736
Pakkanen MS, 2011, Brownian semistationary processes and conditional full support, International Journal of Theoretical and Applied Finance, Vol: 14, Pages: 579-586, ISSN: 0219-0249
Pakkanen MS, 2010, Stochastic Integrals and Conditional Full Support, Journal of Applied Probability, Vol: 47, Pages: 650-667, ISSN: 0021-9002
<jats:p>We present conditions that imply the conditional full support (CFS) property, introduced in Guasoni, Rásonyi and Schachermayer (2008), for processes <jats:italic>Z</jats:italic> := <jats:italic>H</jats:italic> + ∫<jats:italic>K</jats:italic> d<jats:italic>W</jats:italic>, where <jats:italic>W</jats:italic> is a Brownian motion, <jats:italic>H</jats:italic> is a continuous process, and processes <jats:italic>H</jats:italic> and <jats:italic>K</jats:italic> are either progressive or independent of <jats:italic>W</jats:italic>. Moreover, in the latter case, under an additional assumption that <jats:italic>K</jats:italic> is of finite variation, we present conditions under which <jats:italic>Z</jats:italic> has CFS also when <jats:italic>W</jats:italic> is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.</jats:p>
Pakkanen MS, 2010, Microfoundations for diffusion price processes, Mathematics and Financial Economics, Vol: 3, Pages: 89-114, ISSN: 1862-9679
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